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\documentclass[lang=cn,10pt,a4paper,twocolumn]{elegantpaper}
\usepackage{float}
\usepackage{wrapfig}
\title{半导体物理和微电子器件公式整理}
\author{yezoli}
\date{}
\begin{document}

\maketitle
\section{能带}
$E_g(t) = E_g(0) - \frac{\alpha T^2}{\beta + T}$\\

$m^* = \frac{\hbar^2}{\frac{d^2E}{dk^2}}$\\

$m^*_n = -m^*_p$\\

$g_c(E) = \frac{dZ}{dE} = \frac{V}{2\pi^2}\frac{(2m^*_n)^\frac{3}{2}}{\hbar^3}(E-E_c)^\frac{1}{2}$\\

$g_v(E) = \frac{dZ}{dE} = \frac{V}{2\pi^2}\frac{(2m^*_p)^\frac{3}{2}}{\hbar^3}(E_v-E)^\frac{1}{2}$\\

$n_0p_0 = n_i^2$\\

$f_{Fn}(E) = \frac{1}{1 + e^{\frac{E - E_F}{k_0T}}}$\\

$f_{Bn}(E) = e^{-\frac{E - E_F}{k_0T}}$\\

$N_c = 2(\frac{2\pi m_n^*k_0T}{\hbar^2})^\frac{3}{2}$\\

$n_0 = N_ce^{\frac{E_F - E_c}{k_0T}}$\\

$N_v = 2(\frac{2\pi m_p^*k_0T}{\hbar^2})^\frac{3}{2}$\\

$p_0 = N_ve^{\frac{E_v - E_F}{k_0T}}$\\

$n_i = \sqrt{N_cN_v}e^{-\frac{E_g}{2k_0T}}$\\

$E_Fi = E_i + k_0Tln(\frac{N_v}{N_c})$\\

$f_{D}(E) = \frac{1}{1 + \frac{1}{2}e^{\frac{E_D - E_F}{k_0T}}}$\\

$f_{A}(E) = \frac{1}{1 + \frac{1}{4}e^{\frac{E_F - E_A}{k_0T}}}$\\
\section{非平衡}
\subsection{复合}
$\Delta n(t) = \Delta n_0e^{\frac{t}{r}}$\\

$R = rnp$\\

$G_0 = R_0 = rn_i^2$\\

$U_0 = R - G_0 = rnp - rn_i^2 = r(n_0 + p_0)\Delta p + r(\Delta p)^2$\\

$U_0 = \frac{np - n_i^2}{\tau(n + p + 2n_i)}$ \\

$U_0 = \frac{\Delta p}{\tau}$\\

$\tau = \frac{1}{r[(n_0 + p_0) + \Delta p]}$\\

\subsection{漂移}
$\mu_n = \frac{q\tau_n}{m^*_n}$\\

电离杂质：$\tau \propto \frac{T^{\frac{3}{2}}}{N_i}$\\

声学波晶格振动：$\tau \propto T^{-\frac{3}{2}}$\\

光学波晶格振动：$\tau \propto (e^{\frac{h\nu}{k_0T}} - 1)$\\

$\frac{1}{\tau} = \sum \frac{1}{\tau_i}$\\

$\mu_n = \frac{q}{m^*_n}(a\frac{N_i}{T^{\frac{3}{2}}} + bT^{\frac{3}{2}})^{-1}$\\

$\sigma = q(\Delta p\mu_p + \Delta n\mu_n)$\\

$v_d = \frac{\mu_nE}{(1 + (\frac{\mu_nE}{v_{sat}})^a)^{\frac{1}{a}}}$
\subsection{扩散}
弱场$L_p = \sqrt{D_p * \tau}$\\

强场$L_p(E) = E\mu_p\tau$\\

$J = q(D_n \frac{dn}{dx} - D_p \frac{dp}{dx})$\\

$D_nq = k_0T\mu_n$
\section{金半接触}
\subsection{热电子发射}
$A^* = \frac{qm_n^*k_0^2}{2\pi^2\hbar^3}$\\

$A = 120A/cm^2\cdot K^2$\\

$J_{sT} = A^*T^2e^{-\frac{q\phi_{ns}}{k_0T}}$\\

$J = J_{sT}(e^{\frac{qV}{k_0T}} - 1)$\

\subsection{扩散}
$x_d = \sqrt{-\frac{2\varepsilon_r\varepsilon_0(V_s + V)}{qN_D}}$\\

$\sigma = qn_0\mu_{n0}$\\

$J_{sD} = \sigma\sqrt{-\frac{2qN_D}{\varepsilon_r\varepsilon_0}(V_s + V)}e^{-\frac{qV_{D}}{k_0T}}= \frac{q^2D_nN_c}{k_0T}\sqrt{-\frac{2qN_d}{\varepsilon_r\varepsilon_0}(V_s + V)}e^{-\frac{q\phi_{ns}}{k_0T}}$\\

$J = J_{sD}(e^{\frac{qV}{k_0T}} - 1)$\\

\subsection{电容}
$E(x) = \frac{qN_D}{\varepsilon_r\varepsilon_0}(x - x_d)$\\

$V(x) = \frac{qN_D}{\varepsilon_r\varepsilon_0}(xx_d - \frac{1}{2}x^2) - \phi_{ns}$\\

$C = \frac{\varepsilon_r\varepsilon_0}{x_d}$\\

\subsection{镜像力}
$V(x) = \frac{q}{16\pi\varepsilon_r\varepsilon_0x} + \frac{qN_D}{\varepsilon_r\varepsilon_0}(xx_d - \frac{1}{2}x^2) - \phi_{ns}$\\

$x_m = \frac{1}{4\sqrt{\pi N_d x_d}}$\\

$q\Delta\phi = \frac{q}{4}[\frac{2q^7N_D}{\pi^2\varepsilon_r^3\varepsilon_0^3}(V_D -V)]^\frac{1}{4}$\\

\section{MIS}
泊松方程: $\frac{d^2V}{dx^2} = -\frac{\rho}{\varepsilon_{rs}\varepsilon_0}$\\

F函数: $F(\frac{qV}{k_0T}, \frac{n_{p0}}{p_{p0}}) = \sqrt{(e^{-\frac{qV}{k_0T}} + \frac{qV}{k_0T} - 1) + \frac{n_{p0}}{p_{p0}}(e^{\frac{qV}{k_0T}} - \frac{qV}{k_0T} - 1)}$\\

德拜长度$L_D$: $\sqrt{\frac{\varepsilon_{rs}\varepsilon_0k_0T}{q^2p_{p0}}}$\\

$E = \frac{\sqrt{2}k_0T}{qL_D}F(\frac{qV}{k_0T}, \frac{n_{p0}}{p_{p0}})$\\

$Q_s = -\varepsilon_{rs}\varepsilon_0E_s$\\

$C_s = \frac{\varepsilon_{rs}\varepsilon_0}{\sqrt{2}L_D}\frac{-e^{-\frac{qV}{k_0T}} + 1 +  \frac{n_{p0}}{p_{p0}}(e^{\frac{qV}{k_0T}} - 1)}{F(\frac{qV_s}{k_0T}, \frac{n_{p0}}{p_{p0}})}$\\
\subsection{多子积累}
$F(\frac{qV}{k_0T}, \frac{n_{p0}}{p_{p0}}) = e^{-\frac{qV}{2k_0T}}$\\

$E_s = -\frac{\sqrt{2}k_0T}{qL_D}e^{-\frac{qV_s}{2k_0T}}$\\

$Q_s =\frac{\sqrt{2}\varepsilon_{rs}\varepsilon_0k_0T}{qL_D}e^{-\frac{qV_s}{2k_0T}}$\\

$C_s = \frac{\varepsilon_{rs}\varepsilon_0}{\sqrt{2}L_D}e^{-\frac{qV_s}{2k_0T}}$\\

\subsection{平带}

$C_s = \frac{\varepsilon_{rs}\varepsilon_0}{L_D}\frac{1 - \frac{qV_s}{2k_0T} + \frac{n_{p0}}{p_{p0}}(1 + \frac{qV_s}{2k_0T})}{\sqrt{1 + \frac{n_{p0}}{p_{p0}}}} \approx \frac{\varepsilon_{rs}\varepsilon_0}{L_D}$\\

\subsection{耗尽 or 临界强反型}
$F(\frac{qV}{k_0T}, \frac{n_{p0}}{p_{p0}}) = \sqrt{\frac{qV_S}{k_0T}}$\\

$E_s = \frac{\sqrt{2}}{L_D}\sqrt{\frac{k_0TV_s}{q}}$\\

$Q_s = -\frac{\sqrt{2}\varepsilon_{rs}\varepsilon_0}{L_D}\sqrt{\frac{k_0TV_s}{q}}$\\

$C_s = \frac{\varepsilon_{rs}\varepsilon_0}{\sqrt{2}L_D}\sqrt{\frac{k_0T}{qV_s}}$\\

$x_d = \frac{\varepsilon_{rs}\varepsilon_0}{C_s}$\\

\subsection{反型}
表面少子浓度$n_s = n_{p_0}e^{\frac{qV_s}{k_0T}} = \frac{n_i^2}{p_{p0}}e^{\frac{qV_s}{k_0T}}$

$F(\frac{qV}{k_0T}, \frac{n_{p0}}{p_{p0}}) = \sqrt{\frac{n_{p0}}{p_{p_0}}}e^{\frac{qV}{2k_0T}}$\\

$E_s = \sqrt{n_s\frac{2k_0T}{\varepsilon_{rs}\varepsilon_0}}$\\

$Q_s = -\sqrt{2k_0Tn_s\varepsilon_{rs}\varepsilon_0}$\\

$C_s = \frac{\varepsilon_{rs}\varepsilon_0}{\sqrt{2}L_D}\sqrt{\frac{n_s}{p_{p0}}}$\\

$x_d = \sqrt{\frac{4\varepsilon_{rs}\varepsilon_0V_B}{qN_A}} = \sqrt{\frac{4\varepsilon_{rs}\varepsilon_0k_0T}{q^2N_A}ln(\frac{N_A}{n_i})}$\\

\subsection{电容}
$C = \frac{1}{\frac{1}{C_0} + \frac{1}{C_s}}$

\subsection{开启电压}
$V_t = 2V_B - \frac{Q_s}{C_o} + V_{FB} = \frac{2k_0T}{q}ln(\frac{N_A}{n_i}) - \frac{Q_s}{C_o} + \frac{W_m - W_s}{q} - \frac{1}{C_o}\int_0^{d_0}\frac{x\rho(x)}{d_0}dx$

\section{PN结}
$V_{bi}=\frac{kT}{q}\ln{\frac{N_{A}N_{D}}{n_i^2}}$ \\

突变结：

$|E_{max}|=\sqrt{\frac{2qN_0}{\varepsilon_s}V_{bi}}$ \\

$x_n=\frac{\varepsilon_s}{qN_D}|E_{max}|$ \\

$x_p=\frac{\varepsilon_s}{qN_A}|E_{max}|$ \\

$x_d = \sqrt{\frac{2\varepsilon_s}{qN_0}V_{bi}}$ \\

缓变结：

$|E_{max}|=\frac{1}{8}(\frac{aq}{\varepsilon_s})^\frac{1}{3}(12V_{bi})^\frac{2}{3}$ \\

$x_d = \sqrt{\frac{8\varepsilon_sE_{max}}{aq}} = \sqrt[3]{\frac{12\varepsilon_sV_{bi}}{aq}}$ \\

\subsection{电流}
$p_{n} = p_{n0}e^{\frac{qV}{kT}}$ \\

$J_d = J_{dp} + J_{dn} = q(\frac{D_pp_{n0}}{L_p} + \frac{D_nn_{p0}}{L_n})(e^{\frac{qV}{kt}}-1)$ \\

$J_d = J_{dp} + J_{dn} = q(\frac{D_pp_{n0}}{W_p} + \frac{D_nn_{p0}}{W_n})(e^{\frac{qV}{kt}}-1)$ 薄基区二极管\\

$J_{gr} = \frac{qn_ix_d}{2\tau}\frac{e^{\frac{qV}{kT}}-1}{e^{\frac{qV}{2kT}}+1}$ \\

$p_n = p_p = n_n = n_p = n_ie^{\frac{qV}{2kT}}$ 大注入 \\

$V_{KN} = \frac{2kT}{q}ln(\frac{\sqrt{2}N_D}{n_i})$ 大注入 \\

$J_d = J_{dp} + J_{dn} = q(\frac{\sqrt{2}D_pp_{ni}}{L_p} + \frac{\sqrt{2}D_nn_{ni}}{L_n})(e^{\frac{qV}{kt}}-1)$ 大注入 \\

\subsection{击穿}
\subsection{雪崩击穿}
$V_B = \frac{\varepsilon_sE_C^2}{2qN_0} = 5.2*10^{13}E_G^{\frac{3}{2}}N_0^{-\frac{3}{4}}$ 突变结\\

$V_B = \sqrt{\frac{32\varepsilon_s}{9aq}}E_C^{\frac{3}{2}} = 10^{10}E_G^{\frac{6}{5}}a^{-\frac{2}{5}}$ 缓变结\\

\subsection{齐纳击穿}
$d = \frac{E_G}{q|E_{max}|}$

\subsection{电容}
\subsection{势垒电容}
$C_T = \sqrt{\frac{\varepsilon_sqN_0}{2(V_{bi}-V)}}$ 突变结\\

$C_T = \sqrt[3]{\frac{aq\varepsilon_s}{12(V_{bi}-V)}}$ 缓变结\\

\subsection{扩散电容}
$C_D = \frac{qI\tau}{2kT}$ \\

\section{晶体管}
\subsection{突变结}
$I_E=I_{pE}=I_{nE}$\\

$I_B=I_{nE}=I_{nr}$ \\

$I_C=I_{pC}=I_{pE}-I_{pr}=I_E-I_{nE}-I_{nr}$ \\

$\alpha=\frac{I_C}{I_E} | _{V_{EB}>0,V_{CB}=0}$ \\

$h_{FB}=\frac{I_C}{I_E} | _{V_{EB}>0,V_{CB}<0}$ \\

$\beta=\frac{I_C}{I_B} | _{V_{EB}>0,V_{CB}=0}$ \\

$h_{FE}=\frac{I_C}{I_B} | _{V_{EB}>0,V_{CB}<0}$ \\

$\beta = \frac{\alpha}{1-\alpha}$\\

$\alpha = \frac{\beta}{1+\beta}$\\

$\alpha_{0}=\frac{dI_C}{dI_E} | _{V_{EB}>0,V_{CB}<0}$ \\

$\beta_{0}=\frac{dI_C}{dI_B} | _{V_{EB}>0,V_{CB}<0}$ \\

$\beta^*=\frac{I_{pC}}{I_{pE}}=\frac{J_{pC}}{J_{pE}}$ \\

$\beta_B(x)=p_B(0)\frac{sinh[(W_B-x)/L_B]}{sinh(W_B/L_B)}$ \\

$J_{pE}=\frac{qD_Bp_B(0)cosh(W_B/L_B)}{L_Bsinh(W_B/L_B)}$ \\

$J_{pC}=\frac{qD_Bp_B(0)}{L_Bsinh(W_B/L_B)}$ \\

$\beta^*=sinh(\frac{W_B}{L_B}) = 1-\frac{1}{2}(\frac{W_B}{L_B})^2$ \\

$Q_B=\frac{qp_B(0)W_B}{2}$ \\

$\tau_b=\frac{Q_B}{J_{pE}}=\frac{W_B^2}{2D_B}$ \\

$\tau_B=\frac{Q_B}{J_{pr}}$\\

$\beta^*=1-\frac{\tau_b}{\tau_B}$ \\

$\gamma=\frac{I_{pE}}{I_E}=\frac{I_{pE}}{I_{pE}+I_{nE}}=\frac{1}{1+\frac{nE}{pE}}=\frac{1}{1+\frac{D_EW_BN_B}{D_BW_EN_E}}=1-\frac{D_EW_BN_B}{D_BW_EN_E}=1-\frac{W_B\rho_E}{W_E\rho_B}=1-\frac{R_{\Box E}}{R_{\Box B1}}$ \\

$\alpha = 1-\frac{W_B^2}{2L_B^2}-\frac{R_{\Box R}}{R_{\Box B1}} = 1-\delta$ \\

$\beta=\frac{1-\delta}{\delta}\approx \delta^{-1}=(\frac{W_B^2}{2L_B^2}+\frac{R_{\Box E}}{R_{\Box B1}})^{-1}$ \\

\subsection{缓变结}
$N_B(x)=N_B(0)e^{-\frac{\eta x}{W_B}}$ \\

$\eta=ln\frac{N_B(0)}{N_B(W_B)}$ \\

$E=\frac{D_ndp_B(x)}{\mu_np_B(x)dx}$ \\

$E=\frac{D_ndN_B(x)}{\mu_nN_B(x)dx}=-\frac{\eta kT}{qW_B}$ 小注入\\

$J_{nE}=\frac{qD_nn_i^2}{\int_0^{W_b}N_Bdx}[e^{\frac{qV_{BE}}{kT}}-1]$

$n_B(x)=\frac{J_{nE}W_B(1-e^{-\eta(1-x/W_B)})}{qD_n\eta}$ \\

$lim_{\eta -> 0}n_B(x)=n_B(0)(1-\frac{x}{W_B})$ \\

$\tau_b=\frac{q\int_0^{W_B}n_B(x)dx}{J_{nE}}=\frac{2W_B^2}{2D_n\eta}(1-\frac{1}{\eta}+\frac{e^{-\eta}}{\eta})$ \\

$\beta^* = 1-\frac{W_B^2}{L_B^2\eta}(1-\frac{1}{\eta})$ \\

$J_{nE} = \frac{qD_nn_i^2}{\int_0^{W_B}N_Bdx}(e^{\frac{qV_{BE}}{kT}}-1)= qkT\mu_p\mu_nR_{\Box B1}n_i^2(e^{\frac{qV_{BE}}{kT}}-1)$ \\

$\gamma=1-\frac{R_{\Box E}}{R_{\Box B1}}$ \\

$\alpha=\beta^*\gamma\approx 1-\frac{W_B^2}{L_B^2\eta}(1-\frac{1}{\eta})-\frac{R_{\Box R}}{R_{\Box B1}}=1-\delta$ \\

$\beta=\frac{\alpha}{1-\alpha}=\delta^{-1}=[\frac{W_B^2}{L_B^2 \eta}(1-\frac{1}{\eta})+\frac{R_{\Box E}}{R_{\Box B1}}]^{-1}$ \\ 

$\gamma=\frac{1}{1+\frac{R_{\Box E}}{R_{\Box B1}}+\frac{J_{rE}}{J_{nE}}}$ 不忽略$J_{rE}$\\

$\frac{J_{rE}}{J_{nE}}=\frac{x_dN_E}{2L_Bn_i}e^{-\frac{qV_{BE}}{2kT}}$ \\

发射区重掺杂：

发射区禁带宽度变窄$\Delta E_G=\frac{3q}{16\pi\epsilon_s}(\frac{q^2N_E}{\epsilon_skT})^{\frac{1}{2}}$ \\

$\gamma = 1 - \frac{R_{\Box E}}{R_{\Box B1}}e^{\frac{\Delta E_G}{kT}}$ \\

\subsection{直流电流电压}
\subsubsection{集电结短路}
$I_{nE}=A_E \frac{qD_Bn_i^2}{\int_0^{W_B}N_Bdx}(e^{\frac{qV_{BE}}{kT}}-1)$ \\

$I_{pE}=A_E \frac{qD_En_i^2}{\int_0^{W_E}N_Rdx}(e^{\frac{qV_{BE}}{kT}}-1)$ \\

$I_{E}=I_{nE} + I_{pE}=A_EkT\mu_n\mu_pn_i^2(R_{\Box B1}+R_{\Box E})(e^{\frac{qV_{BE}}{kT}}-1)=I_{ES}(e^{\frac{qV_{BE}}{kT}}-1) = I_{ES}(e^{\frac{qV_{BE}}{kT}}-1)$ \\

$I_{C}=\alpha I_{E}$ \\

$I_{B}=I_E-I_C$ \\

\subsubsection{发射结短路}
$I_{E}=\alpha_R I_{C}$ \\

$I_{C}=-I_{CS}(e^{\frac{qV_{BC}}{kT}}-1)$ \\

$I_{B}=(1-\alpha_R)I_C$ \\

\subsubsection{E-M方程}
$I_{E}=I_{ES}(e^{\frac{qV_{BE}}{kT}}-1)-\alpha_R I_{CS}(e^{\frac{qV_{BC}}{kT}}-1)$ \\

$I_{C}=\alpha I_{ES}(e^{\frac{qV_{BE}}{kT}}-1)-I_{CS}(e^{\frac{qV_{BC}}{kT}}-1)$ \\

$\alpha_R I_{CS}=\alpha I_{ES}$ \\

\subsubsection{共基极}
$I_{CBO}=(1-\alpha\alpha_R)I_{CS}$ \\

$I_{C}=\alpha I_{E}-I_{CBO}(e^{\frac{qV_{BC}}{kT}}-1)$ \\

\subsubsection{共射极}
$I_{CEO}=\frac{1-\alpha\alpha_R}{1-\alpha}I_{CS}$ \\

$I_{C}=\beta I_{B}-I_{CEO}(e^{\frac{q(V_{BE}-V_{CE})}{kT}}-1)$ \\

\subsection{厄利效应}
$I_{C}=I_{nE}=A_E \frac{qD_Bn_i^2}{\int_0^{W_B}N_Bdx}(e^{\frac{qV_{BE}}{kT}}-1)$ \\

$\frac{\partial I_{C}}{\partial V_{CE}}|_{V_{BE}}=A_EqD_Bn_i^2(e^{\frac{qV_{BE}}{kT}}-1)(-\frac{N_B(W_B)\frac{dW_B}{dV_{CE}}}{(\int_0^{W_B}N_Bdx)^2})=I_C(-\frac{N_B(W_B)\frac{dW_B}{dV_{CE}}}{(\int_0^{W_B}N_Bdx)^2})\equiv\frac{I_C}{V_A}\equiv\frac{1}{r_o}$ \\

$V_A=\frac{\int_0^{W_B}N_Bdx}{N_B(W_B)\frac{dx_{dB}}{dV_{CB}}|_{V_{CB}=0}}$ \\

$r_o=\frac{V_A}{I_C}$ \\

$x_{dB}=(\frac{2\epsilon_s N_C(V_{bi}+V_{CB})}{N_B(N_B+N_C)})^{\frac{1}{2}}$ \\

\subsection{反向特性}
$I_{ES}=I_{E}|V_{BE}<0,V_{BC}=0$ \\

$I_{CS}=I_{C}|V_{BE}=0,V_{BC}<0$ \\

$I_{CBO}=I_{C}|I_{E}=0,V_{BC}<0$ \\

$I_{CEO}=I_{E}|I_{B}=0,V_{BC}<0$ \\

$I_{EBO}=I_{E}|I_{C}=0,V_{BE}<0$ \\

$I_{CBO}->V_{BE}=\frac{kT}{q}ln(1-\alpha)<0$ \\

$I_{CBO}=(1-\alpha\alpha_R)I_{CS}<I_{CS}<I_{CEO}=\frac{1-\alpha\alpha_R}{1-\alpha}I_{CS}$ \\

$BV_{CEO}\sqrt[S]{\beta}=BV_{CBO}$
\subsection{雪崩击穿}
$M=\frac{1}{1-(frac{|V|}{V_B})^S}$ \\

$I_C=\alpha I_E + I_{CBO} -> I_C=\alpha MI_E+MI_{CBO}$ 共基级\\

$I_C=\frac{\alpha}{1-\alpha} I_B + \frac{I_{CEO}}{1-\alpha} -> I_C= \frac{\alpha M}{1-\alpha M} I_B + \frac{M I_{CEO}}{1-\alpha M}$ 共射级\\

$BV_{CEO}=\frac{BV_{CBO}}{s\surd{\beta}}$ \\

\subsection{基区穿通效应}
$x_{dB}=(\frac{2\epsilon_sN_CV_{pt}}{qN_B(N_C+N_B)})^{\frac{1}{2}}=W_B$ \\

$V_{pt}=\frac{q}{2\epsilon_s}\frac{N_B}{N_C}(N_C+N_B)W_B^2$ \\

$BV_{CBO}=V_{pt}+BV_{EBO}$ \\

$BV_{CEO}=V_{pt}+V_F\approx V_{pt}$ \\

\subsection{基级电阻}
$r_{bb'}=r_{con}+r_{cb}+r_b+r_{b'}$ \\

$R_{\Box}=\frac{\rho}{W}=\frac{1}{\delta W}=\frac{1}{q\mu NW}=\frac{1}{q\mu\int_0^WNdx}$ \\

$r_{con}=\frac{C_\Omega}{A}$ \\

$r_{b}=\frac{Length}{Width}R_{\Box B2}$ \\

$r_{cb}=\frac{S_b}{6l}R_{\Box B3}$ \\

$r_{b'}=\frac{S_e}{12l}R_{\Box B1}$ \\

\subsection{频率}
$\beta_\omega^*=\frac{\beta_0^*}{1+j\omega\tau_b}=\frac{1-\tau_b/\tau_B}{1+j\omega\tau_b}$ \\

$\beta_\omega^*=\frac{\beta_0^*}{\sqrt{1+\omega^2+\tau_b^2}}e^{-j\omega\tau_b}$ \\

$m=\frac{\tau_{dB}}{\tau_b-\tau_{dB}}$ \\

$\beta_\omega^*=\frac{\beta_0^*}{1+j\omega\tau'_b}e^{-j\omega\frac{m}{1+m}\tau_b}=\frac{\beta_0^*}{1+j\omega\tau'_b}e^{-j\omega m\tau'_b}$ \\

$r_e=\frac{dv_{EB}}{di_E}=\frac{v_{eb}}{i_e}=\frac{kT}{qI_E}$ \\

$\tau_{eb}=r_eC_{TE}$ \\

$\gamma_\omega=\frac{\gamma_0}{1+j\omega\tau_{eb}}=\frac{1-\frac{R_{\Box E}}{R_{\Box B1}}}{1+j\omega\tau_{eb}}$ \\

$\tau_{b}=\frac{C_{DE}v_{eb}}{i_{pc}}=C_{DE}r_e$ \\

$\frac{i_{pcc}}{i_{pc}}=\frac{1}{1+j\omega\frac{\tau_t}{2}}=\frac{1}{1+j\omega\tau_d}$ \\

$\tau_d=\frac{\tau_t}{2}=\frac{x_{dc}}{2v_{max}}$ \\

$\frac{i_{c}}{i_{pcc}}=\frac{1}{1+j\omega C_{TC}r_{cs}}=\frac{1}{1+j\omega\tau_c}$ \\

$\tau_c=C_{TC}r_{cs}$ \\

$\alpha_\omega=\frac{\alpha_0e^{-j\omega m\tau'_b}}{1+j\omega(\tau_ec-m\tau'_b)}=\frac{\alpha_0e^{-j\omega m\tau_{ee}}}{[1+\omega^2(\tau_ec-m\tau'_b)^2]^{\frac{1}{2}}}$ \\

$\beta_\omega=\frac{\beta_0}{1+j\frac{\omega}{\omega_\beta}}=\frac{\beta_0}{1+j\frac{f}{f_\beta}}$ \\

$f_T=\beta_0f_\beta=\frac{1}{2\pi\tau_{ec}}=\frac{1}{2\pi(\tau_{eb}+\tau_{b}+\tau_{d}+\tau_{c})}$ \\

$I_b = (\frac{1}{r_\pi}+j\omega C_\pi)V_{be}-(\frac{1}{r_\mu}+j\omega C_\mu)V_{cb}$ \\

$I_c = g_mV_{be}+(\frac{1}{r_o}+j\omega C_{\mu})V_{cb}$ \\

$K_{pmax}=\frac{P_{0max}}{P_{in}}=\frac{f_T}{8\pi r_{bb'}C_{TC}f^2}$ \\

$M = K_{pmax}f^2=\frac{f_T}{8\pi r_{bb'}C_{TC}}$ \\

$f_M=\sqrt{\frac{f_T}{8\pi r_{bb'}C_{TC}}}=\sqrt{M}$ \\

\subsection{模型}

$r_{\pi}=\beta_0\frac{kT}{qI_E}$ \\

$C_{\pi}=C_{DE}+C_{TE}$ \\

$r_{\mu}=\beta_{0}\frac{V_A}{I_C}$ \\

$C_{\mu}=\frac{r_e}{r_o}C_{DE}+C_{TE}=\frac{kTI_C}{qI_EV_A}C_{DE}+C_{TE}$ \\

$g_{m}=\frac{qI_C}{kT}$ \\

$r_{o}=\frac{V_A}{I_C}$ \\

\section{MOEFET}
$V_T=V_B + V_{FB}+\frac{2qN_A\epsilon_s(2\phi_{FP}+V_S-V_B)^\frac{1}{2}}{C_{OX}}+2\phi_{FP}+V_S-V_B=\phi_{MS}-\frac{Q_{OX}}{C_{OX}}+\frac{2qN_A\epsilon_s(2\phi_{FP}+V_S-V_B)^\frac{1}{2}}{C_{OX}}+2\phi_{FP}+V_S$ \\

$V_T=\phi_{MS}-\frac{T_{OX}Q_{OX}}{\epsilon_{OX}}-\frac{T_{OX}Q_{AD}}{\epsilon_{OX}}+2\phi_{FP}(V_S=0,V_B=0)$ \\

$\Delta V_T=-\frac{T_{OX}Q_{1}}{\epsilon_{OX}}-\frac{T_{OX}Q_{A}}{\epsilon_{OX}}(\sqrt{1-\frac{qN_1R^2}{4\epsilon_s\phi_{FP}}}-1)$ 离子注入 \\

$\Delta V_T=K\sqrt{2\phi_{FP}}(\sqrt{1-\frac{V_{BS}}{2\phi_{FP}}}-1)=\frac{2qN_A\epsilon_s}{C_{OX}}\sqrt{2\phi_{FP}}(\sqrt{1-\frac{V_{BS}}{2\phi_{FP}}}-1)$ 体效应 \\
\subsection{直流电压电流}
非饱和 \\

$I_D=-\frac{Z}{L}\mu_n\int_{V_S}^{V_D}Q_ndV$ \\

$Q_n=-C_{OX}(V_G-V_B-V_{FB}-\phi_{S,inv})-Q_A$ \\

$\phi_{S,inv}=2\phi_{FP}-V_B+V(y)$ \\

$x_d(y)=\sqrt{\frac{2\epsilon_s\phi_{S,inv}(y)}{qN_A}}$  \\

$Q_A(y)=-qN_Ax_d=-\sqrt{2\epsilon_sqN_A[2\phi_{FP}-V_B+V(y)]}$ \\

$Q_n=-C_{OX}(V_G-V_{FB}-2\phi_{FP}-V(y))+\sqrt{2\epsilon_sqN_A[2\phi_{FP}-V_B+V(y)]}$ \\

$I_D=\frac{Z}{L}\mu_nC_{OX}\{(V_G-V_{FB}-2\phi_{FP})(V_D-V_S)-\frac{1}{2}(V_D^2-V_S^2)-\frac{2}{3}\frac{\sqrt{\epsilon_sqN_A}}{C_{OX}}[(2\phi_{FP}-V_B+V_D)^\frac{3}{2}-(2\phi_{FP}-V_B+V_S)^\frac{3}{2}]\}$ \\

$Q_n\approx -C_{OX}[V_{GS}-V_T-V(y)]$ \\

$I_D\approx \frac{Z}{L}\mu_nC_{OX}[(V_{GS}-V_T)V_{DS}-\frac{1}{2}V_{DS}^2]$ \\

饱和 \\

$\Delta L=\sqrt{\frac{2\epsilon_s(V_{DS}-V{Dsat})}{qN_A}}$ \\

$I_D\approx \frac{Z}{2(L-\Delta L)}\mu_nC_{OX}(V_{GS}-V_T)^2$ \\

亚阈值 \\

$I_D= \frac{Z}{L}\mu_n(\frac{kT}{q})^2C_{D}e^{\frac{q}{kT}\frac{V_{GS}-V_T}{1+\frac{C_D}{C_{OX}}}}(1-e^{-\frac{qV_{DS}}{kT}})$
\subsection{电势}
\section{常数}
$n_i(Si, 300K) = 1.43e10/cm^3$\\

$n_i(Ge, 300K) = 2.4e13/cm^3$ \\

$n_i(GeAs, 300K) = 1.8e6/cm^3$ \\

$E_g(Si) = 1.12eV$\\

$E_g(Ge) = 0.67eV$\\

$E_g(GeAs) = 1.4eV$\\

$E_g(GaN) = 3.39eV$\\
\end{document}
